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Full text of "Treatise On Analysis Vol-Ii"

9 THE PLANCHEREL-GODEMENT THEOREM 375 (1 5.9.2.9) Definition ofmF. Let us denote by <D ^^£($3) the set of all continuous functions F which tend to 0 at infinity and for which there exists a bounded measure mF on Sg satisfying (15.9.2.8) for all x, y in A. We shall prove that <3> contains tfc($g)y by means of the following three lemmas. (1 5.9.2.1 0) <D is an ideal in the algebra ^(Sg), It is clear that <D is a vector space. Also if F e $ and G e ^(S^), it follows from (15.9.2.8) that (GF) • mXty = (G*f) • mF = (jtf) - (G • mF), so that GF e <3>, and we may take mGF = G • mF (because G is wF-integ- rable (13.20.5)). (15.9.2.11) For each function FeO, there exists only one bounded measure mF satisfying (15.9.2.8) for all x, y in A. The mapping Fh->/% is linear, and mF ^ 0 whenever F §: 0. To prove the uniqueness of WF , it is enough to show that the functions of the form xj> (where x, y e A) form a total set in the Banach space ^(S^), or again that the functions of the form A + jep (A e C) form a total set in the Banach space ^C(S^)- Now, from the fact that A is an algebra and x\-+x is an algebra homomorphism, it follows that the set B of these functions is a sub- algebra of #c(Sp» containing the constants. This subalgebra separates the points of S^. For if ft, #2 are two distinct points of Sg, then by virtue of (ii) there exists x e A such that xfa) ^ x(x2)- Hence either j£2(ft) ^ .*2fe) or •#3fe) 7* ^3(X2). On the other hand, if S; = S^ u {0}, then for each x ¥* 0 there exists x e A such that x(%) ^ 0, and therefore x2(%) ^ 0, which proves our assertion. Finally, since 5 = (x*)~ (15.9.1), the conjugate of each function in B belongs to B. Hence we may apply the Stone-Weierstrass theorem (7.3.2), which establishes the uniqueness of mF . The linearity of the mapping FH-»/% follows immediately from the uniqueness just established. It remains now to show that the relation F J2; 0 implies that mF > 0. Let G JS> 0 be a function belonging to ^fR(S^). By virtue of (ii), there exists a sequence (*„) of elements of A such that the functions xn converge uniformly on S^ to the function G1/2. The functions xnxn therefore converge uniformly on S^ to G, and hence, by virtue of (15.9.2.8) = lim t\ n-+oo Jrmation is an isometry of stg onto ^c(X(*aQ) (15.4.14); the